% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 2000.
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\centerline{\Largebf ACCOUNT OF THE ICOSIAN CALCULUS}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Proceedings of the Royal Irish Academy,
6 (1858), p.\ 415--416.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 2000}
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\centerline{{\largeit Account of the Icosian Calculus.}}
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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
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\centerline{Communicated November~10, 1856.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~6 (1858), p.\ 415--416.]}
\bigskip
Sir William Rowan Hamilton read a Paper on a new System of Roots
of Unity, and of operations therewith connected: to which system
of symbols and operations, in consequence of the geometrical
character of some of their leading interpretations, he is
disposed to give the name of the `{\sc Icosian Calculus}.'
This Calculus {\it agrees\/} with that of the Quaternions, in
three important respects: namely, 1st, that its three chief
symbols, $\iota$,~$\kappa$,~$\lambda$, are (as above suggested)
{\it roots\/} of {\it unity}, as $i$,~$j$,~$k$ are certain
{\it fourth roots\/} thereof: 2nd, that these new roots
{\it obey\/} the {\it associative\/} law of multiplication; and
3rd, that they are {\it not\/} subject to the {\it commutative\/}
law, or that their {\it places\/} as {\it factors\/} must
{\it not\/} in general be {\it altered\/} in a {\it product}.
And it {\it differs\/} from the Quaternion Calculus, 1st, by
involving roots with {\it different exponents\/}; and 2nd by
{\it not requiring\/} so far as yet appears) the
{\it distributive\/} property of multiplication. In fact, $+$
and $-$, in these new calculuations, enter {\it only\/} as
{\it connecting exponents}, and {\it not\/} as connecting
{\it terms\/}: indeed, {\it no terms}, or in other words,
{\it no polynomes}, nor even binomes, have hitherto presented
themselves, in these late researches of the author. As regards
the {\it exponents\/} of the new roots, it may be mentioned that
in the {\it principal system\/}---for the new Calculus involves a
{\it family of systems\/}---there are adopted the equations,
$$1 = \iota^2 = \kappa^3 = \lambda^5,\quad
\lambda = \iota \kappa;
\eqno ({\rm A})$$
so that we deal, in it, with a {\it new square root, cube root},
and {\it fifth root}, of {\it positive unity\/}; the latter root
being the {\it product\/} of the two former, when taken in an
{\it order\/} assigned, but {\it not\/} in the opposite order.
From these simple assumptions~(A), a long train of consistent
calculations opens itself out, for every result of which there is
found a corresponding geometrical interpretation, in the theory
of two of the celebrated solids of antiquity, alluded to with
interest by Plato in the Timaeus; namely, the Icosaedron, and the
Dodecaedron: whereof the {\it angles\/} may {\it now\/} be
{\it unequal}. By making $\lambda^4 = 1$, the author obtains
other symbolical results, which are interpreted by the Octahedron
and the Hexahedron. The Pyramid is, in {\it this\/} theory,
almost too simple to be interesting: but it is dealt with by the
assumption, $\lambda^3 = 1$, the other equations~(A) being
untouched. As one fundamental result of those equations~(A),
which may serve as a slight specimen of the rest, it is found
that if we make $\iota \kappa^2 = \mu$, we shall have
$$\mu^5 = 1,\quad
\mu = \lambda \iota \lambda,\quad
\lambda = \mu \iota \mu;$$
so that this {\it new fifth root\/}~$\mu$ has relations of
perfect {\it reciprocity\/} with the former fifth root~$\lambda$.
But there exist more {\it general\/} results, {\it including\/}
this, and others, on which Sir W.~R.~H. hopes to be allowed to
make a future communication to the Academy: as also on some
applications of the principles already stated, or alluded to,
which appear to be in some degree interesting.
\bye